arXiv subjects

Felix Hermann

Publications and source records attributed to Felix Hermann.

7 recordsLinked to original sources

arXiv2026

Logarithmic scaling of selective sweep curves: from tents to houses

One of the classical results of mathematical population genetics states that the frequency of a beneficial mutant's offspring, on its way to fixation in a large population, looks like a logistic curve. A logarithmic scaling (both in height and time) of these selective sweep curves leads (in the case of strong selection) to a tent-like shape in the large population limit: First the logarithmic frequency of the mutant increases linearly from 0 to 1, then that of the former resident decreases from 1 to 0. For moderate selection the logarithmic frequencies develop (in the large population limit) a jump at the beginning/the end of the sweep, which takes the shape of the tent into that of a house. Our main result (proved for the Moran model) assesses the regularity of this convergence in the large population limit: It is uniform in the house's roof (phases of linear growth and decline) and "Skorokhod $M_1$" in the house's walls (closely around the jumps). Apart from interest in its own right, we anticipate that this result and the proof techniques will be instrumental for extending the description of clonal interference by Poissonian interacting trajectories (as it was done in Hermann et al. (2024) for strong selection) also to moderate selection.

math.PR↗

arXiv2025

The Offended Voter Model

We study a variant of the voter model on a coevolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability $q$. Otherwise, with probability $1-q$, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of "segregation'', otherwise of "consensus''. We rigorously show that segregation and a weaker form of consensus both occur with positive probability for every $q \in (0,1)$ and that the segregation probability tends to $1$ as $q \to 0$. Furthermore, we establish that, if $q \to 1$ fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. We provide results from simulations to assess the obtained bounds and to discuss further properties of the limiting state.

math.PR↗

arXiv2024

From clonal interference to Poissonian interacting trajectories

We consider a population whose size $N$ is fixed over the generations, and in which random beneficial mutations arrive at a rate of order $1/\log N$ per generation. In this so-called Gerrish--Lenski regime, typically a finite number of contending mutations are present together with one resident type. These mutations compete for fixation, a phenomenon addressed as clonal interference. We introduce and study a Poissonian system of interacting trajectories (PIT), and prove that it arises as a large population scaling limit of the logarithmic sizes of the contending clonal subpopulations in a continuous-time Moran model with strong selection. We show that the PIT exhibits an almost surely positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation. We relate this speed to heuristic predictions from the literature. Furthermore, we derive a functional central limit theorem for the fitness of the resident population in the PIT.

math.PR↗

arXiv2022

The Contact Process with Switching

In this paper, we introduce a type switching mechanism for the Contact Process on the lattice $\mathbb{Z}^d$. That is, we allow the individual particles/sites to switch between two (or more) types independently of one another, and the different types may exhibit specific infection and recovery dynamics. Such type switches can eg.\ be motivated from biology, where 'phenotypic switching' is common among micro-organisms. Our framework includes as special cases systems with switches between 'active' and 'dormant' states (the Contact Process with dormancy, CPD), and the Contact Process in a randomly evolving environment (CPREE) introduced by Broman (2007). The 'standard' multi-type Contact Process (without type-switching) can also be recovered as a limiting case. After constructing the process from a graphical representation, we establish several basic properties that are mostly analogous to the classical Contact Process. We then provide couplings between several variants of the system, which in particular yield the existence of a phase transition. Further, we investigate the effect of the switching parameters on the critical value of the system by providing rigorous bounds obtained from the coupling arguments as well as numerical and heuristic results. Finally, we investigate scaling limits for the process as the switching parameters tend to 0 (slow switching regime) resp.\ $\infty$ (fast switching regime). We conclude with a brief discussion of further model variants and questions for future research.

math.PR↗

arXiv2020

A branching process model for dormancy and seed banks in randomly fluctuating environments

The goal of this article is to contribute towards the conceptual and quantitative understanding of the evolutionary benefits for (microbial) populations to maintain a seed bank (consisting of dormant individuals) when facing fluctuating environmental conditions. To this end, we compare the long term behaviour of `1-type' Bienaym\'e-Galton-Watson branching processes (describing populations consisting of `active' individuals only) with that of a class of `2-type' branching processes, describing populations consisting of `active' and `dormant' individuals. All processes are embedded in an environment changing randomly between `harsh' and `healthy' conditions, affecting the reproductive behaviour of the populations accordingly. For the 2-type branching processes, we consider several different switching regimes between active and dormant states. We also impose overall resource limitations which incorporate the potentially different `production costs' of active and dormant offspring, leading to the notion of `fair comparison' between different populations, and allow for a reproductive trade-off due to the maintenance of the dormancy trait. Our switching regimes include the case where switches from active to dormant states and vice versa happen randomly, irrespective of the state of the environment (`spontaneous switching'), but also the case where switches are triggered by the environment (`responsive switching'), as well as combined strategies. It turns out that there are rather natural scenarios under which either switching strategy can be super-critical, while the others, as well as complete absence of a seed bank, are strictly sub-critical, even under `fair comparison' wrt. available resources. In such a case, we see a clear selective advantage of the super-critical strategy, which is retained even under the presence of a (potentially small) reproductive trade-off. [...]

math.PR↗

arXiv2019

The partial duplication random graph with edge deletion

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if $p$ is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques.

math.PR↗

arXiv2014

Large-scale behavior of the partial duplication random graph

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$ vertices, $n=n_0,n_0+1,...$ In state $G_n = (V_n, E_n)$, a vertex $v\in V_n$ is chosen from $V_n$ uniformly at random and is partially duplicated. Upon such an event, a new vertex $v'\notin V_n$ is created and every edge $\{v,w\} \in E_n$ is copied with probability~$p$, i.e.\ $E_{n+1}$ has an edge $\{v',w\}$ with probability~$p$, independently of all other edges. Within this graph, we study several aspects for large~$n$. (i) The frequency of isolated vertices converges to~1 if $p\leq p^* \approx 0.567143$, the unique solution of $pe^p=1$. (ii) The number $C_k$ of $k$-cliques behaves like $n^{kp^{k-1}}$ in the sense that $n^{-kp^{k-1}}C_k$ converges against a non-trivial limit, if the starting graph has at least one $k$-clique. In particular, the average degree of a vertex (which equals the number of edges -- or 2-cliques -- divided by the size of the graph) converges to $0$ iff $p<0.5$ and we obtain that the transitivity ratio of the random graph is of the order $n^{-2p(1-p)}$. (iii) The evolution of the degrees of the vertices in the initial graph can be described explicitly. Here, we obtain the full distribution as well as convergence results.

math.PR↗